Editing Euclidean geometry (section)

==The ''Elements''==
{{main|Euclid's Elements}}
The ''Elements'' is mainly a systematization of earlier knowledge of geometry. Its improvement over earlier treatments was rapidly recognized, with the result that there was little interest in preserving the earlier ones, and they are now nearly all lost.

There are 13 books in the ''Elements'':

Books I–IV and VI discuss plane geometry. Many results about plane figures are proved, for example, "In any triangle, two angles taken together in any manner are less than two right angles." (Book I proposition 17) and the [[Pythagorean theorem]] "In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle." (Book I, proposition 47)

Books V and VII–X deal with [[number theory]], with numbers treated geometrically as lengths of line segments or areas of surface regions. Notions such as [[prime numbers]] and [[rational number|rational]] and [[irrational number]]s are introduced. It is proved that there are infinitely many prime numbers.

Books XI–XIII concern [[solid geometry]]. A typical result is the 1:3 ratio between the volume of a cone and a cylinder with the same height and base. The [[platonic solid]]s are constructed.

===Axioms===
[[File:Parallel postulate en.svg|thumb|The parallel postulate (Postulate 5): If two lines intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.]]

Euclidean geometry is an [[axiomatic system]], in which all [[theorem]]s ("true statements") are derived from a small number of simple axioms. Until the advent of [[non-Euclidean geometry]], these axioms were considered to be obviously true in the physical world, so that all the theorems would be equally true. However, Euclid's reasoning from assumptions to conclusions remains valid independently from the physical reality.<ref name=Wolfe>The assumptions of Euclid are discussed from a modern perspective in
{{cite book |title=Introduction to Non-Euclidean Geometry |author=Harold E. Wolfe |url=https://books.google.com/books?id=VPHn3MutWhQC&pg=PA9 |page=9 |isbn=978-1-4067-1852-2 |year=2007 |publisher=Mill Press}}
</ref>

Near the beginning of the first book of the ''Elements'', Euclid gives five [[postulate]]s (axioms) for plane geometry, stated in terms of constructions (as translated by Thomas Heath):<ref>tr. Heath, pp. 195–202.</ref>

:Let the following be postulated:
# To draw a [[straight line]] from any [[Point (geometry)|point]] to any point.
# To produce (extend) a [[Line segment|finite straight line]] continuously in a straight line.
# To describe a [[circle]] with any centre and distance (radius).
# That all [[right angle]]s are equal to one another.
# [The [[parallel postulate]]]: That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than two right angles.

Although Euclid explicitly only asserts the existence of the constructed objects, in his reasoning he also implicitly assumes them to be unique.

The ''Elements'' also include the following five "{{Visible anchor|common notions}}":

# Things that are equal to the same thing are also equal to one another (the [[transitive property]] of a [[Euclidean relation]]).
# If equals are added to equals, then the wholes are equal (Addition property of equality).
# If equals are subtracted from equals, then the differences are equal (subtraction property of equality).
# Things that coincide with one another are equal to one another (reflexive property).
# The whole is greater than the part.

Modern scholars agree that Euclid's postulates do not provide the complete logical foundation that Euclid required for his presentation.<ref>{{citation|first=Gerard A.|last=Venema|title=Foundations of Geometry|year=2006|publisher=Prentice-Hall|page=8|isbn=978-0-13-143700-5}}.</ref> Modern [[Foundations of geometry|treatments]] use more extensive and complete sets of axioms.

===Parallel postulate===
{{main|Parallel postulate}}
To the ancients, the parallel postulate seemed less obvious than the others. They aspired to create a system of absolutely certain propositions, and to them, it seemed as if the parallel line postulate required proof from simpler statements. It is now known that such a proof is impossible since one can construct consistent systems of geometry (obeying the other axioms) in which the parallel postulate is true, and others in which it is false.<ref>{{Citation|title=History of the Parallel Postulate|journal=The American Mathematical Monthly|volume=27|issue=1|pages=16–23|date=Jan 1920|author=Florence P. Lewis|doi=10.2307/2973238|publisher=The American Mathematical Monthly, Vol. 27, No. 1|postscript=.|jstor=2973238}}</ref> Euclid himself seems to have considered it as being qualitatively different from the others, as evidenced by the organization of the ''Elements'': his first 28 propositions are those that can be proved without it.

Many alternative axioms can be formulated which are [[logical equivalence|logically equivalent]] to the parallel postulate (in the context of the other axioms). For example, [[Playfair's axiom]] states:

:In a [[Plane (geometry)|plane]], through a point not on a given straight line, at most one line can be drawn that never meets the given line.

The "at most" clause is all that is needed since it can be proved from the remaining axioms that at least one parallel line exists.

[[File:euclid-proof.svg|thumb|A proof from Euclid's ''Elements'' that, given a line segment, one may construct an equilateral triangle that includes the segment as one of its sides: an equilateral triangle ΑΒΓ is made by drawing circles Δ and Ε centered on the points Α and Β, and taking one intersection of the circles as the third vertex of the triangle.]]

===Methods of proof===
Euclidean Geometry is ''[[Constructive proof|constructive]]''. Postulates 1, 2, 3, and 5 assert the existence and uniqueness of certain geometric figures, and these assertions are of a constructive nature: that is, we are not only told that certain things exist, but are also given methods for creating them with no more than a [[compass and straightedge|compass and an unmarked straightedge]].<ref>Ball, p.&nbsp;56.</ref> In this sense, Euclidean geometry is more concrete than many modern axiomatic systems such as [[set theory]], which often assert the existence of objects without saying how to construct them, or even assert the existence of objects that cannot be constructed within the theory.<ref name=set_theory>Within Euclid's assumptions, it is quite easy to give a formula for area of triangles and squares. However, in a more general context like set theory, it is not as easy to prove that the area of a square is the sum of areas of its pieces, for example. See [[Lebesgue measure]] and [[Banach–Tarski paradox]].</ref> Strictly speaking, the lines on paper are ''[[Scientific modelling|models]]'' of the objects defined within the formal system, rather than instances of those objects. For example, a Euclidean straight line has no width, but any real drawn line will have. Though nearly all modern mathematicians consider [[Non-constructive proof|nonconstructive proofs]] just as sound as constructive ones, they are often considered less [[Mathematical beauty|elegant]], intuitive, or practically useful. Euclid's constructive proofs often supplanted fallacious nonconstructive ones, e.g. some Pythagorean proofs that assumed all numbers are rational, usually requiring a statement such as "Find the greatest common measure of ..."<ref>{{cite book|author=Daniel Shanks|title=Solved and Unsolved Problems in Number Theory|year=2002|publisher=American Mathematical Society}}</ref>

Euclid often used [[proof by contradiction]].<ref>{{cite journal |title=On the Status of Proofs by Contradiction in the Seventeenth Century |first=Paolo |last=Mancosu |journal=Synthese |year=1991 |volume=88 |number=1 |pages=15–41 |doi=10.1007/BF00540091 |jstor=20116923 }}</ref>